Dirichlet’s Theorem on Primes in Arithmetical Progressions

Abstract

The arithmetic progression of odd numbers 1, 3, 5, ... , 2n + 1,... contains infinitely many primes. It is natural to ask whether other arithmetic progressions have this property. An arithmetic progression with first term h and common difference k consists of all numbers of the form

$$kn = h,n = 0,1,2,...,$$

(1)

If h and k have a common factor d, each term of the progression is divisible by d and there can be no more than one prime in the progression if d > 1. In other words, a necessary condition for the existence of infinitely many primes in the arithmetic progression (1) is that (h, k) — 1. Dirichlet was the first to prove that this condition is also sufficient. That is, if (h, k) = 1 the arithmetic progression (1) contains infinitely many primes. This result, now known as Dirichlet’s theorem, will be proved in this chapter.